Distributive abstract logics and the Esakia duality

TL;DR

This paper introduces a general method to connect abstract logics with lattice structures, establishing dualities with topological spaces, specifically focusing on the duality between intuitionistic logics and Esakia spaces.

Contribution

It develops a broad process to translate abstract logics into lattice frameworks, enabling duality results with topological categories, especially for intuitionistic logic and Esakia spaces.

Findings

Established duality between abstract intuitionistic logics and Esakia spaces

Provided a general method for translating abstract logics into lattice structures

Connected logic categories with topological space categories

Abstract

In this paper we develop an almost general process to switch from abstract logics in the sense of Brown and Suszko to lattices. With this method we can establish dualities between some categories of abstract logics to the correspondent topological space categories. In more detail we will explain the duality between the category of abstract intuitionistic logics with intuitionistic morphisms and the category of Esakia spaces with the Esakia morphisms.